Question

Given ${\log }_{a}2=x$, ${\log }_{a}3=y$ and ${\log }_{a}5=z$. Find the value in each of the following in terms of $x$, $y$ and $z$.
(i) ${\log }_{a}15$
(ii) ${\log }_{a}8$
(iii) ${\log }_{a}30$
(iv) ${\log }_a\left(\frac{27}{125}\right)$
(v) ${\log }_a\left(3\frac{1}{3}\right)$
(vi) ${\log }_{a}1.5$

Answer 1

(i) ${\log }_{a}15={\log }_{a}3+{\log }_{a}5=y+z$

(ii) ${\log }_{a}8={\log }_a{2}^3=3{\log }_{a}2=3x$

(iii) ${\log }_{a}30={\log }_a(2\times 3\times 5)={\log }_{a}2+{\log }_{a}3+{\log }_{a}5=x+y+z$

(iv) ${\log }_a\left(\frac{27}{125}\right)={\log }_a{\left(\frac{3}{5}\right)}^3=3{\log }_{a}3-3{\log }_{a}5=3(y-z)$

(v) ${\log }_{a}2+{\log }_{a}5-{\log }_{a}3=x+z-y$

(vi) ${\log }_a\left(\frac{3}{2}\right)={\log }_{a}3-{\log }_{a}2=y-x$